Berthé, V. (Valérie) 1957
Overview
Works:  28 works in 86 publications in 2 languages and 1,344 library holdings 

Genres:  Conference papers and proceedings 
Roles:  Editor, Author, Other, 956, Opponent, Thesis advisor, 958 
Publication Timeline
.
Most widely held works by
V Berthé
Substitutions in dynamics, arithmetics, and combinatorics by
N Pytheas Fogg(
Book
)
16 editions published between 2002 and 2003 in English and held by 498 WorldCat member libraries worldwide
A certain category of infinite strings of letters on a finite alphabet is presented here, chosen among the 'simplest' possible one may build, both because they are very deterministic and because they are built by simple rules (a letter is replaced by a word, a sequence is produced by iteration). These substitutive sequences have a surprisingly rich structure. The authors describe the concepts of quantity of natural interactions, with combinatorics on words, ergodic theory, linear algebra, spectral theory, geometry of tilings, theoretical computer science, diophantine approximation, trancendence, graph theory. This volume fulfils the need for a reference on the basic definitions and theorems, as well as for a stateoftheart survey of the more difficult and unsolved problems
16 editions published between 2002 and 2003 in English and held by 498 WorldCat member libraries worldwide
A certain category of infinite strings of letters on a finite alphabet is presented here, chosen among the 'simplest' possible one may build, both because they are very deterministic and because they are built by simple rules (a letter is replaced by a word, a sequence is produced by iteration). These substitutive sequences have a surprisingly rich structure. The authors describe the concepts of quantity of natural interactions, with combinatorics on words, ergodic theory, linear algebra, spectral theory, geometry of tilings, theoretical computer science, diophantine approximation, trancendence, graph theory. This volume fulfils the need for a reference on the basic definitions and theorems, as well as for a stateoftheart survey of the more difficult and unsolved problems
Combinatorics, automata, and number theory by
V Berthé(
Book
)
20 editions published between 2007 and 2010 in English and held by 386 WorldCat member libraries worldwide
This collaborative volume presents recent trends arising from the fruitful interaction between combinatorics on words, automata and number theory
20 editions published between 2007 and 2010 in English and held by 386 WorldCat member libraries worldwide
This collaborative volume presents recent trends arising from the fruitful interaction between combinatorics on words, automata and number theory
Combinatorics, words and symbolic dynamics by
V Berthé(
Book
)
11 editions published between 2015 and 2016 in English and held by 268 WorldCat member libraries worldwide
"Internationally recognised researchers look at developing trends in combinatorics with applications in the study of words and in symbolic dynamics. They explain the important concepts, providing a clear exposition of some recent results, and emphasise the emerging connections between these different fields. Topics include combinatorics on words, pattern avoidance, graph theory, tilings and theory of computation, multidimensional subshifts, discrete dynamical systems, ergodic theory, numeration systems, dynamical arithmetics, automata theory and synchronised words, analytic combinatorics, continued fractions and probabilistic models. Each topic is presented in a way that links it to the main themes, but then they are also extended to repetitions in words, similarity relations, cellular automata, friezes and Dynkin diagrams. The book will appeal to graduate students, research mathematicians and computer scientists working in combinatorics, theory of computation, number theory, symbolic dynamics, tilings and stringology. It will also interest biologists using text algorithms"
11 editions published between 2015 and 2016 in English and held by 268 WorldCat member libraries worldwide
"Internationally recognised researchers look at developing trends in combinatorics with applications in the study of words and in symbolic dynamics. They explain the important concepts, providing a clear exposition of some recent results, and emphasise the emerging connections between these different fields. Topics include combinatorics on words, pattern avoidance, graph theory, tilings and theory of computation, multidimensional subshifts, discrete dynamical systems, ergodic theory, numeration systems, dynamical arithmetics, automata theory and synchronised words, analytic combinatorics, continued fractions and probabilistic models. Each topic is presented in a way that links it to the main themes, but then they are also extended to repetitions in words, similarity relations, cellular automata, friezes and Dynkin diagrams. The book will appeal to graduate students, research mathematicians and computer scientists working in combinatorics, theory of computation, number theory, symbolic dynamics, tilings and stringology. It will also interest biologists using text algorithms"
Sequences, Groups, and Number Theory(
)
8 editions published in 2018 in English and held by 136 WorldCat member libraries worldwide
This collaborative book presents recent trends on the study of sequences, including combinatorics on words and symbolic dynamics, and new interdisciplinary links to group theory and number theory. Other chapters branch out from those areas into subfields of theoretical computer science, such as complexity theory and theory of automata. The book is built around four general themes: number theory and sequences, word combinatorics, normal numbers, and group theory. Those topics are rounded out by investigations into automatic and regular sequences, tilings and theory of computation, discrete dynamical systems, ergodic theory, numeration systems, automaton semigroups, and amenable groups. This volume is intended for use by graduate students or research mathematicians, as well as computer scientists who are working in automata theory and formal language theory. With its organization around unified themes, it would also be appropriate as a supplemental text for graduate level courses.
8 editions published in 2018 in English and held by 136 WorldCat member libraries worldwide
This collaborative book presents recent trends on the study of sequences, including combinatorics on words and symbolic dynamics, and new interdisciplinary links to group theory and number theory. Other chapters branch out from those areas into subfields of theoretical computer science, such as complexity theory and theory of automata. The book is built around four general themes: number theory and sequences, word combinatorics, normal numbers, and group theory. Those topics are rounded out by investigations into automatic and regular sequences, tilings and theory of computation, discrete dynamical systems, ergodic theory, numeration systems, automaton semigroups, and amenable groups. This volume is intended for use by graduate students or research mathematicians, as well as computer scientists who are working in automata theory and formal language theory. With its organization around unified themes, it would also be appropriate as a supplemental text for graduate level courses.
Actes des Journées Montoises d'Informatique Théorique : 911 Septembre 2002, Montpellier, France by Journées Montoises d'Informatique Théorique(
Book
)
3 editions published in 2003 in French and English and held by 24 WorldCat member libraries worldwide
3 editions published in 2003 in French and English and held by 24 WorldCat member libraries worldwide
Mots de retours et pavages dans les plans sturmiens by
Matthieu Simonet(
)
1 edition published in 2012 in French and held by 3 WorldCat member libraries worldwide
Sturmian words are a way to encode aperiodic discrete lines. They have been studied since the end of the 19th century and can be characterized in many ways. One of these characterizations, obtained by Vuillon, centers around the notion of return words.This thesis aims to study 2dimensional Sturmian words as encodings of aperiodic discrete planes. It is a first step towards a characterization of 2dimensional Sturmian words analogous to that of Vuillon in dimension 1.However, concerns specific to dimension 2, such as the impossibility to concatenate words or the difficulty to locate a factor inside a word make the study much trickier. To tackle this, we introduce in dimension 2 notions of patterns, pointed patterns, localization words and return words.We obtain a 2dimensional version of a theorem of Morse and Hedlund concerning certain return words in a Sturmian word. This result enables us to establish a new continuedfractions algorithm and to introduce, in a restricted setting, a notion of derived sequence
1 edition published in 2012 in French and held by 3 WorldCat member libraries worldwide
Sturmian words are a way to encode aperiodic discrete lines. They have been studied since the end of the 19th century and can be characterized in many ways. One of these characterizations, obtained by Vuillon, centers around the notion of return words.This thesis aims to study 2dimensional Sturmian words as encodings of aperiodic discrete planes. It is a first step towards a characterization of 2dimensional Sturmian words analogous to that of Vuillon in dimension 1.However, concerns specific to dimension 2, such as the impossibility to concatenate words or the difficulty to locate a factor inside a word make the study much trickier. To tackle this, we introduce in dimension 2 notions of patterns, pointed patterns, localization words and return words.We obtain a 2dimensional version of a theorem of Morse and Hedlund concerning certain return words in a Sturmian word. This result enables us to establish a new continuedfractions algorithm and to introduce, in a restricted setting, a notion of derived sequence
Actes des Journées Montoises d'Informatique Théorique : 911 Septembre 2002, Montpellier, France by Journées Montoises d'Informatique Théorique(
Book
)
1 edition published in 2003 in English and held by 3 WorldCat member libraries worldwide
1 edition published in 2003 in English and held by 3 WorldCat member libraries worldwide
Arithmetic distributions of convergents arising from JacobiPerron algorithm by
V Berthé(
Book
)
2 editions published in 2004 in English and held by 3 WorldCat member libraries worldwide
2 editions published in 2004 in English and held by 3 WorldCat member libraries worldwide
Structure des pavages, droites discrètes 3D et combinatoire des mots by
Sébastien Labbé(
)
2 editions published in 2012 in French and held by 3 WorldCat member libraries worldwide
This thesis, consisting of a series of articles, considers issues arising in discrete geometry by treating them in terms of combinatorics of words: a powerful and appropriate tool to address them. We use words to represent either a path in S\ZZA2 S or S\ZZA3 S, either to encode the sequence of turns of a path or represent the boundary of a discrete closed figure. Topics covered include tilings of the plane by copies of a polyomino, the notion of palindrome complexity and generation of 3D discrete lines. The first part concerns the tilings of the plane where we study the number of regular tilings of the plane by a tile. It turns out that some polyominoes can tile the plane in two distinct ways with four tiles adjacent. They are called double square. We first demonstrate there are at most two such regular tilings by a tile. Next, we consider two particular families of double squares: Christoffel tiles and Fibonacci tiles. These two families describe the smallest examples of double squares and can be defined from the Christoffel words and the Fibonacci word by rules of substitution and concatenation. The Fibonacci tiles also define a fractal, obtained by the limit of a sequence of closed selfavoiding path, we calculated several statistics including the ratio of the area of the fractal on the area of its convex hull. In the next article, we demonstrate that indecomposable double squares are invariant under a rotation of 180 degrees. This geometric property is equivalent to the fact that their boundary word can be factorized into a product of palindromes. Our proof is based on an exhaustive method of generating double squares. The second part concerns the palindromic complexity  the number of distinct palindrome factors  a proper subject of combinatorics on words. We consider four classes of palindromic complexity arising naturally from the notion of palindromic defect. We characterize words on a twoletter alphabet having a minimal palindromic complexity and we show that the infinite words obtained by codings of rotations on two intervals reach the maximum palindromic complexity. In the third part, we propose a method based on multidimensional continued fraction algorithms for the generation of 3D discrete lines S6Sconnected. Experiments illustrate that the complexity of factors in words and generated would be linear. This compares favorably with other definitions of 3D discrete S6Sconnected lines whose factor complexity is quadratic. Seven articles are included in the thesis
2 editions published in 2012 in French and held by 3 WorldCat member libraries worldwide
This thesis, consisting of a series of articles, considers issues arising in discrete geometry by treating them in terms of combinatorics of words: a powerful and appropriate tool to address them. We use words to represent either a path in S\ZZA2 S or S\ZZA3 S, either to encode the sequence of turns of a path or represent the boundary of a discrete closed figure. Topics covered include tilings of the plane by copies of a polyomino, the notion of palindrome complexity and generation of 3D discrete lines. The first part concerns the tilings of the plane where we study the number of regular tilings of the plane by a tile. It turns out that some polyominoes can tile the plane in two distinct ways with four tiles adjacent. They are called double square. We first demonstrate there are at most two such regular tilings by a tile. Next, we consider two particular families of double squares: Christoffel tiles and Fibonacci tiles. These two families describe the smallest examples of double squares and can be defined from the Christoffel words and the Fibonacci word by rules of substitution and concatenation. The Fibonacci tiles also define a fractal, obtained by the limit of a sequence of closed selfavoiding path, we calculated several statistics including the ratio of the area of the fractal on the area of its convex hull. In the next article, we demonstrate that indecomposable double squares are invariant under a rotation of 180 degrees. This geometric property is equivalent to the fact that their boundary word can be factorized into a product of palindromes. Our proof is based on an exhaustive method of generating double squares. The second part concerns the palindromic complexity  the number of distinct palindrome factors  a proper subject of combinatorics on words. We consider four classes of palindromic complexity arising naturally from the notion of palindromic defect. We characterize words on a twoletter alphabet having a minimal palindromic complexity and we show that the infinite words obtained by codings of rotations on two intervals reach the maximum palindromic complexity. In the third part, we propose a method based on multidimensional continued fraction algorithms for the generation of 3D discrete lines S6Sconnected. Experiments illustrate that the complexity of factors in words and generated would be linear. This compares favorably with other definitions of 3D discrete S6Sconnected lines whose factor complexity is quadratic. Seven articles are included in the thesis
Fonction de Carlitz et automates ; entropies conditionnelles by
V Berthé(
Book
)
2 editions published in 1994 in French and held by 2 WorldCat member libraries worldwide
CE TRAVAIL SE DIVISE EN DEUX PARTIES INDEPENDANTES. DANS LA PREMIERE, NOUS DEMONTRONS, EN UTILISANT LES AUTOMATES FINIS, DES RESULTATS DE TRANSCENDANCE DANS LE MODULE DE CARLITZ DE RANG 1. DANS LA SECONDE PARTIE, NOUS NOUS INTERESSONS A DES ENTROPIES CONDITIONNELLES DE BLOCS, AU SENS DE SHANNON, POUR DES SUITES A VALEURS DANS UN ALPHABET FINI, ET EN PARTICULIER, POUR LES SUITES STURMIENNES
2 editions published in 1994 in French and held by 2 WorldCat member libraries worldwide
CE TRAVAIL SE DIVISE EN DEUX PARTIES INDEPENDANTES. DANS LA PREMIERE, NOUS DEMONTRONS, EN UTILISANT LES AUTOMATES FINIS, DES RESULTATS DE TRANSCENDANCE DANS LE MODULE DE CARLITZ DE RANG 1. DANS LA SECONDE PARTIE, NOUS NOUS INTERESSONS A DES ENTROPIES CONDITIONNELLES DE BLOCS, AU SENS DE SHANNON, POUR DES SUITES A VALEURS DANS UN ALPHABET FINI, ET EN PARTICULIER, POUR LES SUITES STURMIENNES
Purely periodic betaexpansions in the Pisot nonunit case by
V Berthé(
Book
)
3 editions published in 2002 in English and held by 2 WorldCat member libraries worldwide
3 editions published in 2002 in English and held by 2 WorldCat member libraries worldwide
Shift spaces on groups : computability and dynamics by
Sebastián Andrés Barbieri Lemp(
)
1 edition published in 2017 in English and held by 1 WorldCat member library worldwide
Shift spaces are sets of colorings of a group which avoid a set of forbidden patterns and are endowed with a shift action. These spaces appear naturally as discrete versions of dynamical systems: they are obtained by partitioning the phase space and mapping each element into the sequence of partitions visited by its orbit.Severa! breakthroughs in this domain have pointed out the intricate relationship between dynamics of shift spaces and their computability properties. One remarkable example is the classification of the entropies of multidimensional subshifts of finite type as the set of right recursively enumerable numbers. This work explores shift spaces with a dual approach: on the one hand we are interested in their dynamical properties and on the ether hand we studythese abjects as computational models.Four salient results have been obtained as a result of this approach: (1) a combinatorial condition ensuring nonemptiness of subshifts on arbitrary countable groups; (2) a simulation theorem which realizes effective actions of finitely generated groups as factors of a subaction of a subshift of finite type; (3) a characterization of effectiveness with oracles using generalized Turing machines and (4) the undecidability of the torsion problem for two group invariants of shift spaces.As byproducts of these results we obtain a simple proof of the existence of strongly aperiodic subshifts in countable groups. Furthermore, we realize them as subshifts of finite type in the case of a semidirect product of a ddimensional integer lattice with a finitely generated group with decida ble word problem whenever d> 1
1 edition published in 2017 in English and held by 1 WorldCat member library worldwide
Shift spaces are sets of colorings of a group which avoid a set of forbidden patterns and are endowed with a shift action. These spaces appear naturally as discrete versions of dynamical systems: they are obtained by partitioning the phase space and mapping each element into the sequence of partitions visited by its orbit.Severa! breakthroughs in this domain have pointed out the intricate relationship between dynamics of shift spaces and their computability properties. One remarkable example is the classification of the entropies of multidimensional subshifts of finite type as the set of right recursively enumerable numbers. This work explores shift spaces with a dual approach: on the one hand we are interested in their dynamical properties and on the ether hand we studythese abjects as computational models.Four salient results have been obtained as a result of this approach: (1) a combinatorial condition ensuring nonemptiness of subshifts on arbitrary countable groups; (2) a simulation theorem which realizes effective actions of finitely generated groups as factors of a subaction of a subshift of finite type; (3) a characterization of effectiveness with oracles using generalized Turing machines and (4) the undecidability of the torsion problem for two group invariants of shift spaces.As byproducts of these results we obtain a simple proof of the existence of strongly aperiodic subshifts in countable groups. Furthermore, we realize them as subshifts of finite type in the case of a semidirect product of a ddimensional integer lattice with a finitely generated group with decida ble word problem whenever d> 1
Avoidability of Abelian Repetitions in Words by
Matthieu Rosenfeld(
)
1 edition published in 2017 in English and held by 1 WorldCat member library worldwide
In this document, we study the avoidability of different kind of repetitions in words. We firstshow that under some conditions one can decide whether a morphic word avoids abelian nthpowers. This algorithm can decide over a wider class of morphism than the previousalgorithms. We generalize this algorithm and use it to show that long abelian squares areavoidable over the ternary alphabet and that additive squares are avoidable over Z2 . The firstresult answers a weak version of a question formulated by Mäkelä in 2003 and the second oneis related to an open question from 1994 about the avoidability of additive squares over Z.Another generalization of this algorithm can be used to study avoidability of patterns in theabelian sense. In particular, we show that binary patterns of length more than 14 areavoidable over the binary alphabet in the abelian sense. This improves considerably theprevious bound of 118.We give sufficient conditions for a morphism to be long kabelian nth powerfree. This resultallows us to compute for every k ≥ 3 the number of different kabelian squares that a binaryword must contain. We prove that long 2abelian squares are avoidable over the binaryalphabet and that over the ternary alphabet there exists a word that contains only one 2abelian square.We also give a complete classification of binary formulas based on the size of the smallestalphabet over which they are avoidable and on the growth (exponential or polynomial) of theassociated language
1 edition published in 2017 in English and held by 1 WorldCat member library worldwide
In this document, we study the avoidability of different kind of repetitions in words. We firstshow that under some conditions one can decide whether a morphic word avoids abelian nthpowers. This algorithm can decide over a wider class of morphism than the previousalgorithms. We generalize this algorithm and use it to show that long abelian squares areavoidable over the ternary alphabet and that additive squares are avoidable over Z2 . The firstresult answers a weak version of a question formulated by Mäkelä in 2003 and the second oneis related to an open question from 1994 about the avoidability of additive squares over Z.Another generalization of this algorithm can be used to study avoidability of patterns in theabelian sense. In particular, we show that binary patterns of length more than 14 areavoidable over the binary alphabet in the abelian sense. This improves considerably theprevious bound of 118.We give sufficient conditions for a morphism to be long kabelian nth powerfree. This resultallows us to compute for every k ≥ 3 the number of different kabelian squares that a binaryword must contain. We prove that long 2abelian squares are avoidable over the binaryalphabet and that over the ternary alphabet there exists a word that contains only one 2abelian square.We also give a complete classification of binary formulas based on the size of the smallestalphabet over which they are avoidable and on the growth (exponential or polynomial) of theassociated language
Discrétisations spatiales de systèmes dynamiques génériques by PierreAntoine Guihéneuf(
)
1 edition published in 2015 in English and held by 1 WorldCat member library worldwide
How is it possible to read the dynamical properties (ie when the time goes to infinity) of a system on numerical simulations ? To try to answer this question, we study inthis thesis a model reflecting what happens when the orbits of a discrete time system f (for example an homeomorphism) are computed numerically. The computer working in finite numerical precision, it will replace f by a spacial discretization of f, denotedby f_N (where the order N of discretization stands for the numerical accuracy). In particular, we will be interested in the dynamical behaviour of the finite maps f_N for a generic system f and N going to infinity, where generic will be taken in the sense of Baire (mainly among sets of homeomorphisms or C^1diffeomorphisms). The first part of this manuscript is devoted to the study of the dynamics of the discretizations f_N, when f is a generic conservative/dissipative homeomorphism of a compact manifold. We show that it would be mistaken to try to recover the dynamics of f from that of a single discretization f_N : its dynamics strongly depends on the order N. To detect some dynamical features of f we have to consider all thediscretizations f_N when N goes through N.The second part deals with the linear case, which plays an important role in the study of C^1generic diffeomorphisms, discussed in the third part of this manuscript. Under these assumptions, we obtain results similar to those established in the first part,though weaker and harder to prove
1 edition published in 2015 in English and held by 1 WorldCat member library worldwide
How is it possible to read the dynamical properties (ie when the time goes to infinity) of a system on numerical simulations ? To try to answer this question, we study inthis thesis a model reflecting what happens when the orbits of a discrete time system f (for example an homeomorphism) are computed numerically. The computer working in finite numerical precision, it will replace f by a spacial discretization of f, denotedby f_N (where the order N of discretization stands for the numerical accuracy). In particular, we will be interested in the dynamical behaviour of the finite maps f_N for a generic system f and N going to infinity, where generic will be taken in the sense of Baire (mainly among sets of homeomorphisms or C^1diffeomorphisms). The first part of this manuscript is devoted to the study of the dynamics of the discretizations f_N, when f is a generic conservative/dissipative homeomorphism of a compact manifold. We show that it would be mistaken to try to recover the dynamics of f from that of a single discretization f_N : its dynamics strongly depends on the order N. To detect some dynamical features of f we have to consider all thediscretizations f_N when N goes through N.The second part deals with the linear case, which plays an important role in the study of C^1generic diffeomorphisms, discussed in the third part of this manuscript. Under these assumptions, we obtain results similar to those established in the first part,though weaker and harder to prove
Probabilistic analyses of the plain multiple gcd algorithm(
)
1 edition published in 2016 in English and held by 1 WorldCat member library worldwide
Abstract: Among multiple gcd algorithms on polynomials as on integers, one of the most natural ones performs a sequence of ℓ − 1 phases ( ℓ is the number of inputs), with each of them being the Euclid algorithm on two entries. We present here a complete probabilistic analysis of this algorithm, by providing both the averagecase and the distributional analysis, and by handling in parallel the integer and the polynomial cases, for polynomials with coefficients in a finite field. The main parameters under consideration are the number of iterations in each phase and the evolution of the size of the current gcd along the execution. Three phenomena are clearly emphasized through this analysis: the fact that almost all the computations are performed during the first phase, the great difference between the probabilistic behavior of the first phase compared to subsequent phases, and, as can be expected, the great similarity between the integer and the polynomial cases
1 edition published in 2016 in English and held by 1 WorldCat member library worldwide
Abstract: Among multiple gcd algorithms on polynomials as on integers, one of the most natural ones performs a sequence of ℓ − 1 phases ( ℓ is the number of inputs), with each of them being the Euclid algorithm on two entries. We present here a complete probabilistic analysis of this algorithm, by providing both the averagecase and the distributional analysis, and by handling in parallel the integer and the polynomial cases, for polynomials with coefficients in a finite field. The main parameters under consideration are the number of iterations in each phase and the evolution of the size of the current gcd along the execution. Three phenomena are clearly emphasized through this analysis: the fact that almost all the computations are performed during the first phase, the great difference between the probabilistic behavior of the first phase compared to subsequent phases, and, as can be expected, the great similarity between the integer and the polynomial cases
Combinatoire de substitutions de type Pisot ombinatorics of Pisot substitutions by
Timothée Jolivet(
Book
)
1 edition published in 2013 in French and held by 1 WorldCat member library worldwide
Substitutions are mappings which replace each symbol of a given alphabet by a word over the same alphabet. They naturally act over infinite sequences of symbols, and produce highly ordered Systems with many properties. This thesis concerns a particular class with algebraic restrictions, Pisot substitutions, and their related objects of dynamical, fractal o combinatorial nature. We begin with the combinatorial study of some qualitative properties of the twodimensional patterns generated by iterating a twodimensional "dual" version of Pisot substitutions. We apply these results to study the infinite families of substitutions obtained by taking arbitrary products over a finite set of Pisot substitutions. Applications include dynamica properties of the associated symbolic Systems, some language theoretical characterization of some topological properties of their associated Rauzy fractals, some numbertheoretical properties of their associated Pisot numbers, and some results in discrete geometry. Particular focus is set on the substitutions associated with the ArnouxRauzy, Brun and JacobiPerron multidimensional continued fraction algorithms Next we give explicit construction to give a complete description of the possible fondamental groups of planar Rauzy fractals in the case where the group is countable. In the last two chapters, we "step back" from the Pisot algebraic assumption to study some more general objects arising from the combinatorial tools used in the previous chapters, focusing on some computational (un)decidability questions
1 edition published in 2013 in French and held by 1 WorldCat member library worldwide
Substitutions are mappings which replace each symbol of a given alphabet by a word over the same alphabet. They naturally act over infinite sequences of symbols, and produce highly ordered Systems with many properties. This thesis concerns a particular class with algebraic restrictions, Pisot substitutions, and their related objects of dynamical, fractal o combinatorial nature. We begin with the combinatorial study of some qualitative properties of the twodimensional patterns generated by iterating a twodimensional "dual" version of Pisot substitutions. We apply these results to study the infinite families of substitutions obtained by taking arbitrary products over a finite set of Pisot substitutions. Applications include dynamica properties of the associated symbolic Systems, some language theoretical characterization of some topological properties of their associated Rauzy fractals, some numbertheoretical properties of their associated Pisot numbers, and some results in discrete geometry. Particular focus is set on the substitutions associated with the ArnouxRauzy, Brun and JacobiPerron multidimensional continued fraction algorithms Next we give explicit construction to give a complete description of the possible fondamental groups of planar Rauzy fractals in the case where the group is countable. In the last two chapters, we "step back" from the Pisot algebraic assumption to study some more general objects arising from the combinatorial tools used in the previous chapters, focusing on some computational (un)decidability questions
Automates Cellulaires : aspects algorithmiqued des configurations périodiques en toute dimension by
Nicolas Bacquey(
Book
)
1 edition published in 2015 in French and held by 1 WorldCat member library worldwide
This thesis analyses the computational capabilities of cellular automata working on periodical configurations of any dimension. We first study the maximal objects these cellular automata can identify; we call those objects primitive roots of periodical configurations of any dimension. We characterize them and show some of their properties. Secondly, we present a set of algorithms on cellular automata, each one adapted to one or more dimensions, that extract primitive roots from the periodical configurations on which they are applied. Those algorithms use original tools that extend the notion of signals on cellular automata. Beyond its technical and algorithmical aspects, this thesis lays the foundations of uniform periodical computation, i.e. computation performed on a model whose program and entry data are isotropic. In particular, we address the issues of halting such computation, reading its result and defining its temporal or spatial complexity
1 edition published in 2015 in French and held by 1 WorldCat member library worldwide
This thesis analyses the computational capabilities of cellular automata working on periodical configurations of any dimension. We first study the maximal objects these cellular automata can identify; we call those objects primitive roots of periodical configurations of any dimension. We characterize them and show some of their properties. Secondly, we present a set of algorithms on cellular automata, each one adapted to one or more dimensions, that extract primitive roots from the periodical configurations on which they are applied. Those algorithms use original tools that extend the notion of signals on cellular automata. Beyond its technical and algorithmical aspects, this thesis lays the foundations of uniform periodical computation, i.e. computation performed on a model whose program and entry data are isotropic. In particular, we address the issues of halting such computation, reading its result and defining its temporal or spatial complexity
Initial powers of Sturmian sequences by
V Berthé(
)
1 edition published in 2006 in English and held by 1 WorldCat member library worldwide
1 edition published in 2006 in English and held by 1 WorldCat member library worldwide
Systèmes dynamiques substitutifs et renormalisation by
Jordan Emme(
)
1 edition published in 2016 in English and held by 1 WorldCat member library worldwide
In the present work we study substitutive dynamical systems. Historically, substitutions have been introduced in order to describe the sequence of the sumofdigits mod 2 in base 2. We study some properties of densities of sets defined by sumofdigits functions, sets which are linked with autocorrelations of some arithmétic functions. We prove that these densities are usually normally distributed. We also study the regularity of the pressure function in the framework of the thermodynamics formalism, introduced by Bowen, Ruelle and Sinaï, for a family of potentials defined in terms of distance to the attractor of the kbonacci substitution. We also show that the iterations of the renormalisation operator defined by Baraviera, Leplaideur and Lopes converges towards a fixed point of this operator. Finally we study the regularity of some spectral measures associated to selfsimilar tilings using mostly works from Bufetov and Solomyak on the deviations of ergodic sums for the action of translations by vectors in R^d on selfsimilar tilings of R^d. We prove that, afeter renormalisation, these spectral measures behave like Radon measures around
1 edition published in 2016 in English and held by 1 WorldCat member library worldwide
In the present work we study substitutive dynamical systems. Historically, substitutions have been introduced in order to describe the sequence of the sumofdigits mod 2 in base 2. We study some properties of densities of sets defined by sumofdigits functions, sets which are linked with autocorrelations of some arithmétic functions. We prove that these densities are usually normally distributed. We also study the regularity of the pressure function in the framework of the thermodynamics formalism, introduced by Bowen, Ruelle and Sinaï, for a family of potentials defined in terms of distance to the attractor of the kbonacci substitution. We also show that the iterations of the renormalisation operator defined by Baraviera, Leplaideur and Lopes converges towards a fixed point of this operator. Finally we study the regularity of some spectral measures associated to selfsimilar tilings using mostly works from Bufetov and Solomyak on the deviations of ergodic sums for the action of translations by vectors in R^d on selfsimilar tilings of R^d. We prove that, afeter renormalisation, these spectral measures behave like Radon measures around
Formal languages, automata and numeration systems by
Michel Rigo(
)
1 edition published in 2014 in English and held by 1 WorldCat member library worldwide
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1 edition published in 2014 in English and held by 1 WorldCat member library worldwide
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Related Identities
 Rigo, Michel Other Author Editor
 Pytheas Fogg, N. Author
 SpringerLink (Service en ligne)
 Société mathématique de Belgique
 Bruyère, Véronique
 Université Paul Valéry
 Durand, Fabien (1969....). Opponent
 Zamboni, Luca
 Vuillon, Laurent (1970....). Opponent Thesis advisor
 Brlek, Srecko 1952 Thesis advisor
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Associated Subjects
Combinatorial analysis Computational complexity Computer programming Computer science Computer scienceMathematics Differentiable dynamical systems Formal languages Group theory Information theory Machine theory Mathematics Morphisms (Mathematics) Number theory Rewriting systems (Computer science) Sequences (Mathematics) Symbolic dynamics
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Alternative Names
Berthé, V.
Berthé, V. 1957
Berthé, Valerie
Berthé, Valérie 1957
Valérie Berthé Frans wiskundige
Valérie Berthé französische Mathematikerin
Valérie Berthé French mathematician
Valérie Berthé matemática francesa
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